The Jacobson radical for analytic crossed products

We characterise the (Jacobson) radical of the analytic crossed product of C_0(X) by the non-negative integers (Z_+), answering a question first raised by Arveson and Josephson in 1969. In fact, we characterise the radical of analytic crossed products of C_0(X) by (Z_+)^d. The radical consists of all elements whose `Fourier coefficients' vanish on the recurrent points of the dynamical system (and the first one is zero). The multi-dimensional version requires a variation of the notion of recurrence, taking into account the various degrees of freedom.Comment: 17 pages; AMS-LaTeX; minor correction

Amalgams of Inverse Semigroups and C*-algebras

An amalgam of inverse semigroups [S,T,U] is full if U contains all of the idempotents of S and T. We show that for a full amalgam [S,T,U], the C*-algebra of the inverse semigroup amaglam of S and T over U is the C*-algebraic amalgam of C*(S) and C*(T) over C*(U). Using this result, we describe certain amalgamated free products of C*-algebras, including finite-dimensional C*-algebras, the Toeplitz algebra, and the Toeplitz C*-algebras of graphs

Cartan Triples

We introduce the class of Cartan triples as a generalization of the notion of a Cartan MASA in a von Neumann algebra. We obtain a one-to-one correspondence between Cartan triples and certain Clifford extensions of inverse semigroups. Moreover, there is a spectral theorem describing bimodules in terms of their support sets in the fundamental inverse semigroup and, as a corollary, an extension of Aoi's theorem to this setting. This context contains that of Fulman's generalization of Cartan MASAs and we discuss his generalization in an appendix.Comment: 37 page

CARTAN TRIPLES

We introduce the class of Cartan triples as a generalization of the notion of a Car- tan MASA in a von Neumann algebra. We obtain a one-to-one correspondence between Cartan triples and certain Clifford extensions of inverse semigroups. Moreover, there is a spectral theorem describing bimodules in terms of their support sets in the fundamental inverse semigroup and, as a corollary, an extension of Aoi’s theorem to this setting. This context contains that of Fulman’s generalization of Cartan MASAs and we discuss his generalization in an appendix

Algebraic Isomorphisms and Spectra of Triangular Limit Algebras: Erratum

We have found an error in the proof of Theorem 4.1 in [2]. This error affects only Theorems 4.1 and 4.3. As we have been unable to find an alternative proof of Theorem 4.1, we do not know if these theorems are true in full generality. Restricted to limit algebras generated by their order-preserving normalizers, the proof is correct. The theorems are known to be true in this case by somewhat different methods [1]

JOINS AND COVERS IN INVERSE SEMIGROUPS AND TIGHT C*-ALGEBRAS

We show Exel’s tight representation of an inverse semigroup can be described in terms of joins and covers in the natural partial order. Using this, we show that the C*-algebra of a finitely aligned category of paths, developed by Spielberg, is the tight C*-algebra of a natural inverse semigroup. This includes as a special case finitely aligned higher-rank graphs: that is, for such a higher-rank graph Ʌ, the tight C* -algebra of the inverse semigroup associated to Ʌ is the same as the C*-algebra of Ʌ