Generalizing earlier results about the set of idempotents in a Banach
algebra, or of self-adjoint idempotents in a $C^*$-algebra, we announce
constructions of nice connecting paths in the connected components of the set
of elements in a Banach algebra, or of self-adjoint elements in a
$C^*$-algebra, that satisfy a given polynomial equation, without multiple
roots. In particular, we will prove that in the Banach algebra case every such
non-central element lies on a complex line, all of whose points satisfy the
given equation. We also formulate open questions.Comment: 8 pdf page

Makai, Jr., E.

Zemánek, Jaroslav

English

arXiv

summary:Generalizing earlier results about the set of idempotents in a Banach algebra, or of self-adjoint idempotents in a $C^*$-algebra, we announce constructions of nice connecting paths in the connected components of the set of elements in a Banach algebra, or of self-adjoint elements in a $C^*$-algebra, that satisfy a given polynomial equation, without multiple roots. In particular, we prove that in the Banach algebra case every such non-central element lies on a complex line, all of whose points satisfy the given equation. We also formulate open questions

Makai, Endre Jr.

Czech digital mathematics library

Nice connecting paths in connected components of sets of algebraic elements in a Banach algebra

Generalizing earlier results about the set of idempotents in a Banach algebra, or of self-adjoint idempotents in a C*-algebra, we announce constructions of nice connecting paths in the connected components of the set of elements in a Banach algebra, or of self-adjoint elements in a C*-algebra, that satisfy a given polynomial equation, without multiple roots. In particular, we prove that in the Banach algebra case every such non-central element lies on a complex line, all of whose points satisfy the given equation. We also formulate open questions. © 2016, Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic

Makai, Endre

Zemánek, J.

Repository of the Academy's Library

arXiv:1601.01505v1  [math.FA]  7 Jan 2016NICE CONNECTING PATHS IN CONNECTEDCOMPONENTS OF SETS OF ALGEBRAICELEMENTS IN A BANACH ALGEBRAEndre Makai, Jr., Budapest and Jaroslav Zema´nek, WarszawaDedicated to the 90th anniversary of Professor Miroslav Fiedler,from two grateful participants in mathematical olympiadsMTA Alfre´d Re´nyi Institute of Mathematics,H-1364 Budapest, Pf. 127, Hungaryhttp://www.renyi.mta.hu/˜makaiInstitute of Mathematics, Polish Academy of Sciences,00-656 Warsaw, S´niadeckich 8, Polandmakai.endre@renyi.mta.hu, zemanek@impan.plAbstract. Generalizing earlier results about the set of idempotents in a Banachalgebra, or of self-adjoint idempotents in a C∗-algebra, we announce constructions ofnice connecting paths in the connected components of the set of elements in a Banachalgebra, or of self-adjoint elements in a C∗-algebra, that satisfy a given polynomialequation, without multiple roots. In particular, we will prove that in the Banachalgebra case every such non-central element lies on a complex line, all of whosepoints satisfy the given equation. We also formulate open questions.2010 Mathematics Subject Classification. Primary: 46H20, Secondary: 46L05.Key words and phrases. Banach algebras, C∗-algebras, (self-adjoint) idempo-tents, connected components of (self-adjoint) algebraic elements, (local) pathwiseconnectedness, similarities, analytic paths, polynomial paths, polygonal paths, cen-tre of a Banach algebra, distance of connected components.1. IntroductionTypeset by AMS-TEX2 E. MAKAI, JR., J. ZEMA´NEKLet A be a unital complex Banach algebra. Sometimes we will assume thatmoreover A is a C∗-algebra.We letE(A) := {a ∈ A | a2 = a}be the set of idempotents of A, andS(A) := {a ∈ A | a2 = a = a∗}be the set of self-adjoint idempotents for the C∗-algebra case.The connected components of E(A) and of S(A) have been investigated by manyauthors. To some of them we will refer later at the respective theorems. An ampleliterature is given in [AMMZ].Letp(λ) :=n∏i=1(λ− λi)be a polynomial over C, with all λi’s distinct. In the C∗-algebra case, when con-sidering self-adjoint elements, we will assume that all λi’s are real. (In fact, ifq(λ) :=∏{(λ− λi) | 1 ≤ i ≤ n, λi ∈ R}, then p(a) = 0 and a = a∗ imply q(a) = 0.Thus below we could use q(λ) rather than p(λ).) The λi’s are fixed throughout thispaper.We writeEp(A) := {a ∈ A | p(a) = 0},andSp(A) := {a ∈ A | p(a) = 0, a = a∗}for the C∗-algebra case. Then E(A) and S(A) are special cases of Ep(A) and Sp(A):namely, for p(λ) := λ(λ− 1).We say that {e1, . . . , en} ⊂ A is a partition of unity, or in the C∗-algebra casethat {e1, . . . , en} ⊂ A is a self-adjoint partition of unity, if{e1, . . . , en} ⊂ E(A), or {e1, . . . , en} ⊂ S(A),and eiej = 0 for 1 ≤ i, j ≤ n and i 6= j,and∑ni=1 ei = 1.NICE CONNECTING PATHS 3The detailed proofs of the statements announced in Section 2 will be publishedin [MZ]. The idea of this development originates from personal conversations ofthe authors at the conference Operator Theory and Applications: Perspectives andChallenges, held in Jurata (Hel), Poland, March 18–28, 2010, and from the 2011lecture by the first named author [Mak].2. TheoremsThe “only if” part of the following Proposition 1 comes from the Riesz decom-position theorem.Proposition 1. Let A be a unital complex Banach algebra (C∗-algebra). Let a ∈ A.Then a ∈ Ep(A) (a ∈ Sp(A)) if and only if there exists a (self-adjoint) partition ofunity {e1, . . . , en} such thata =n∑i=1λiei.In the “only if” part, for a ∈ Ep(A) (for a ∈ Sp(A)) one can choose the ei’s aspolynomials of a, with complex (real) coefficients, which depend only on the λi’s.This representation provides the tool for reducing questions about Ep(A) (aboutSp(A)) to those about E(A) (about S(A)). Of course, for the respective proofs forEp(A) (for Sp(A)) one has to work still substantially. As an illustration, we includea sketch of proof of Theorem 7 in Section 3.The distinctness of the λi’s is essential in order that a should have such a simpleform. For T ∈ A := B(l2 ⊕ l2), having a block matrix form (Tij)2i,j=1, which issubdiagonal (i.e., strictly lower triangular), we have T 2 = 0, but T21 ∈ B(l2) canbe as complicated as an element of B(l2) can be.A path in a topological space X is a continuous map f : [0, 1]→ X . We will saythat f(0), f(1) ∈ X are connected by this path f . By a small abuse of languagewe will also say that f([0, 1]) ⊂ X is a path in X (e.g., for polygonal paths). Atopological space X is pathwise connected if any two of its points are connectedby a path in X . A topological space X is locally pathwise connected if each pointx ∈ X has a base of (not necessarily open) neighbourhoods consisting of pathwiseconnected sets.Theorem 2. Let A be a unital complex Banach algebra and C a connected compo-nent of Ep(A). Then C is a relatively open subset of Ep(A). Further, C is locallypathwise connected via each of the following types of paths:1) similarity via an exponential function, i.e., t 7→ e−ctaect;2) a polynomial path of degree at most three;3) a polygonal path of n segments.4 E. MAKAI, JR., J. ZEMA´NEKFor E(A), relative openness of C was proved by J. Zema´nek [Ze], 1) was provedby J. Zema´nek [Ze], 2) was proved by J. Esterle [Es] and M. Tre´mon [Tr85], 3) wasproved by Z. V. Kovarik [Ko] (cf. also [Ze]).Theorem 3. Under the hypotheses of Theorem 2, C is pathwise connected via eachof the following types of paths:1) similarity via a finite product of exponential functions, i.e., t 7→ e−cmt . . . e−c1taec1t . . . ecmt;2) a polynomial path;3) a polygonal path.In fact, there is a path satisfying 1) and 2) simultaneously.For E(A), 1) was proved by J. Zema´nek [Ze], 2) was proved by J. Esterle [Es]and M. Tre´mon [Tr85], 3) was proved by Z. V. Kovarik [Ko] (cf. also [Ze]), and thelast sentence was proved by [Es] and [Tr85].Problem. Does there exist a uniform bound on the “minimum degree” of thesepolynomial connections, possibly depending on n, valid for all Banach algebras?Does such a bound exist, depending on n and on A (or even on C)? Even the caseof a uniform bound for polynomial connections of idempotents is open, even if weallow dependence of the bound on A (or even on C). For some particular cases, see[Tr85] and [MZ89]. ([Tr95] announced a further partial result, but his proof seemsto be incorrect.)Even the “simplest” case A := B(l2) is open. (The case A =: B(Cn) is solvedpositively by [Tr85], the uniform bound being 3, which is sharp. Here the connectedcomponents of E(A) consist of the projections of the same rank.) For A = B(l2),the connected components of E(A) are {e ∈ A | dimN(e) = α, dimR(e) = β},where 0 ≤ α, β ≤ ℵ0 are cardinalities with α + β = ℵ0, cf. [AMMZ] (N(·) is thenull-space and R(·) is the range). By [MZ89], for min{α, β} < ℵ0, in the respectiveconnected component there exists an at most third degree polynomial path betweenany two elements of that component. But even the case α = β = ℵ0 here is open.Theorem 4. Let A be a unital complex C∗-algebra, and C a connected componentof Sp(A). Then C is a relatively open subset of Sp(A). Further, C is locally path-wise connected by similarities via exponential functions, i.e., t 7→ e−ictaeict, whereadditionally c = c∗.For S(A), Theorem 4 was proved by S. Maeda [Mae] (cf. also [Ze]).Theorem 5. Under the hypotheses of Theorem 4, C is pathwise connected by sim-ilarities via finite products of exponential functions, i.e., t 7→ e−icmt . . . e−ic1taeic1t. . . eicmt, where additionally c1 = c∗1, . . . , cm = c∗m.NICE CONNECTING PATHS 5For S(A), Theorem 5 was proved by S. Maeda [Mae] (cf. also [Ze]).For the C∗-algebra case, the analogues of 2) and 3) from Theorems 2 and 3 arefalse for Sp(A). In fact, already the connected component of S(B(C2))consistingof all rank-one orthogonal projections does not contain any non-constant polyno-mial path. (The connected components of S (B(Cn)) consist of the orthogonalprojections of the same rank.)Theorem 6. Let A be a unital complex Banach algebra (C∗-algebra). Let a ∈Ep(A) (let a ∈ Sp(A)). Then a belongs to the centre of A if and only if its connectedcomponent in Ep(A) (in Sp(A)) is {a}.Theorem 6 for E(A) was proved by J. Zema´nek [Ze], for S(A) by S. Maeda [Mae].In Theorem 6, of course, the “only if” part for Sp(A) follows from the “only if”part for Ep(A).Theorem 7. Let A be a unital complex Banach algebra, and C a connected compo-nent of Ep(A). If C is disjoint from the centre of A, then any element of C belongsto a complex line entirely contained in C. In particular, C is unbounded.For E(A), Theorem 7 was proved by J. Zema´nek [Ze].In the C∗-algebra case even the entire Sp(A) has a distance max{|λi| | 1 ≤ i ≤ n}from 0, so the analogue of Theorem 7 for Sp(A) is false.Theorem 6 and Theorem 7 yield the next Corollary 8.Corollary 8. Let A be a unital complex Banach algebra. Then Ep(A) is a unionof its isolated points and of complex lines. Theorem 9. There exists an explicit constant c(λ1, . . . , λn) > 0 (depending onλ1, . . . , λn ∈ R, and invariant under any map (λ1, . . . , λn) 7→ (a+bλ1, . . . , a+bλn)with a, b ∈ R and b 6= 0) such that the following holds. If A is a unital complex C∗-algebra, and C1, C2 are distinct connected components of Sp(A), then the distanceof C1 and C2 is at least c(λ1, . . . , λn) ·min{|λi − λj | | 1 ≤ i, j ≤ n, i 6= j}.Conjecture. Let A be a unital complex Banach algebra (C∗-algebra) and C1, C2distinct connected components of Ep(A) (of Sp(A)). Then the distance of C1 andC2 is at least min{|λi − λj | | 1 ≤ i, j ≤ n, i 6= j}.For n = 2 this conjecture is equivalent to the statement that this distance forEp(A) := E(A) (for Sp(A) := S(A)) is at least 1, which is due to J. Zema´nek [Ze](due to S. Maeda [Mae]). For n ≥ 3 we do not even know whether this distance forthe Banach algebra case is positive.6 E. MAKAI, JR., J. ZEMA´NEKIf true, this conjecture would be sharp, for any Banach algebra: consider λi · 1and λj · 1.The Conjecture for the case of Sp(A) would follow from the Conjecture in thecase of Ep(A). In fact, different connected components of Sp(A) are subsets ofdifferent connected components of Ep(A), by [BFML], Section 1, Applications, 2),also taking into consideration our Proposition 1 and Theorem 3.3. A proofProof of Theorem 7 from Theorem 3 and Theorem 6. If C is disjoint from the centre,then by Theorem 6 it has more than one elements. Let a0 ∈ C be an arbitraryelement of C, and let a1 ∈ C, with a1 6= a0. Then, by Theorem 3, 3), there existsa non-constant polygonal path connecting a0 to a1 in C. Its first non-constantsegment (counted from a0) is the graph of a non-constant polynomial of degree 1,say ofλ 7→ a0 + bλ, from [0, 1] to C (⊂ Ep(A) ⊂ A).Hence(1) b 6= 0 and we have for all λ ∈ [0, 1] identically p(a0 + bλ) = 0.Then the equation in (1) is a polynomial equation, with coefficients from A and ofdegree at most n, for λ ∈ C. (Attention: here the coefficient of λn is bn, which maybe 0 even for b 6= 0.)We make an indirect assumption. If the polynomial(2) C ∋ λ 7→ p(a0 + bλ) ∈ Awere not identically 0 for all λ ∈ C, then for some λ0 ∈ C we would havep(a0 + bλ0) 6= 0.Then for some continuous linear functional a′ on A we would have〈p(a0 + bλ0), a′〉 6= 0.The polynomial(3) C ∋ λ 7→ 〈p(a0 + bλ), a′〉 ∈ CNICE CONNECTING PATHS 7is a C-valued polynomial on C of degree at most n, which would not vanish atλ0 ∈ C. Hence the polynomial in (3) would have at most n distinct roots.However, by (1) we have that the polynomial in (3) vanishes for all λ ∈ [0, 1]identically. This is a contradiction, showing that our indirect assumption is false.That is, the polynomial in (2) is identically 0 for all λ ∈ C. In other words, forall λ ∈ C we havep(a0 + bλ) = 0, i.e., a0 + bλ ∈ Ep(A),which implies by connectedness of C that for all λ ∈ C we have evena0 + bλ ∈ C.Since by (1) b 6= 0, we see thatC contains a complex line passing through its arbitrary point a0.Acknowledgement. The authors are grateful to the organizers of several con-ferences, where this material took its shape. In particular, to the organizers ofthe conference Operator Theory and Applications: Perspectives and Challenges,in Jurata (Hel), Poland, March 18–28, 2010, of the 6th Linear Algebra Workshop,in Kranjska Gora, Slovenia, May 25–June 1, 2011, and of the Sz.-Nagy CentennialConference, in Szeged, Hungary, June 24–28, 2013.References[AMMZ] B. Aupetit, E. Makai, Jr., M. Mbekhta, J. Zema´nek, The connected components ofthe idempotents in the Calkin algebra, and their liftings, In: Operator Theory andBanach Algebras, Conf. Proc., Rabat (Morocco), April 12–14, 1999 (Ed. M. Chidami,R. Curto, M. Mbekhta, F.-H. Vasilescu, J. Zema´nek), Theta, Bucharest, 2003, 23–30.MR 2004g:46062.[BFML] Z. Boulmaarouf, M. Fernandez Miranda, J.-Ph. Labrousse, An algorithmic approachto orthogonal projections and Moore-Penrose inverses, Numer. Funct. Anal. Optim.18 (1997), 55–63. MR 97m:65105.[Es] J. Esterle, Polynomial connections between projections in Banach algebras, Bull. Lon-don Math. Soc. 15 (1983), 253–254. MR 84g:46069.[Ko] Z. V. Kovarik, Similarity and interpolation between projectors, Acta Sci. Math. (Sze-ged) 39 (1977), 341–351. MR 58#2397.[Mae] S. Maeda, On arcs in the space of projections of a C∗-algebra, Math. Japon. 21(1976), 371–374. MR 56#12900.8 E. MAKAI, JR., J. ZEMA´NEK[Mak] E. Makai, Jr., Algebraic elements in Banach algebras (joint work with J. Zema´nek),In: 6th Linear Algebra Workshop, Kranjska Gora, Slovenia, May 25-June 1, 2011,Book of Abstracts, 26.[MZ89] E. Makai, Jr., J. Zema´nek, On polynomial connections between projections, LinearAlgebra Appl. 126 (1989), 91–94. MR 91m:47026.[MZ] E. Makai, Jr., J. Zema´nek, Manuscript under preparation.[Tr85] M. Tre´mon, Polynoˆmes de degre´ minimum connectant deux projections dans une al-ge`bre de Banach, Linear Algebra Appl. 64 (1985), 115–132. MR 86g:46074.[Tr95] M. Tre´mon, On the degree of polynomials connecting two idempotents of a Banachalgebra, Proc. Roy. Irish Acad. Sect. A 95 (1995), 233–235. MR 99f:46068.[Ze] J. Zema´nek, Idempotents in Banach algebras, Bull. London Math. Soc. 11 (1979),177–183. MR 80h:46073.

Endre Makai

Jaroslav Zemánek

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Nice connecting paths in connected components of sets of algebraic
  elements in a Banach algebra

Nice connecting paths in connected components of sets of algebraic elements in a Banach algebra

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