The algebra of bounded linear operators on ℓp ⊕ ℓq has infinitely many closed ideals

We prove that in the reflexive range 1 < p < q < ∞, the algebra ℒ(ℓp⊕ℓq) of all bounded linear operators on ℓp⊕ℓq has infinitely many closed ideals. This solves a problem raised by A. Pietsch [Operator ideals, Math. Monogr. 16, VEB Deutscher Verlag der Wissenschaften, Berlin 1978, Problem 5.3.3] in his book `Operator ideals'.The first author’s research was supported by NSF grant DMS-1160633. The second author was supported by the 2014 Workshop in Analysis and Probability at Texas A&M University

Shift invariant preduals of &#8467;₁(&#8484;)

The Banach space &#8467;&lt;sub&gt;1&lt;/sub&gt;(&#8484;) admits many non-isomorphic preduals, for example, C(K) for any compact countable space K, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on &#8467;&lt;sub&gt;1&lt;/sub&gt;(&#8484;) weak&lt;sup&gt;*&lt;/sup&gt;-continuous. This is equivalent to making the natural convolution multiplication on &#8467;&lt;sub&gt;1&lt;/sub&gt;(&#8484;) separately weak*-continuous and so turning &#8467;&lt;sub&gt;1&lt;/sub&gt;(&#8484;) into a dual Banach algebra. We call such preduals &lt;i&gt;shift-invariant&lt;/i&gt;. It is known that the only shift-invariant predual arising from the standard duality between C&lt;sub&gt;0&lt;/sub&gt;(K) (for countable locally compact K) and &#8467;&lt;sub&gt;1&lt;/sub&gt;(&#8484;) is c&lt;sub&gt;0&lt;/sub&gt;(&#8484;). We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak&lt;sup&gt;*&lt;/sup&gt;-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to c&lt;sub&gt;0&lt;/sub&gt;. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of &#8484;. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to c&lt;sub&gt;0&lt;/sub&gt;

On The Robustness Of Option Pricing

. In this work we consider the following problem: Let (S (n) t ) 0tT , n 2 N , be a sequence of stochastic processes describing the price of a stock during the time period [0; T ]. We assume that the distribution P n of (S (n) t ) converges weakly to the distribution P of a stochastic process (S t ) 0tT . Secondly, we consider a European style derivative paying F (S) at time T if the stock price at time T is S. For n 2 N let F (n) 0 be an arbitrage free value of that derivative at time 0, assuming the underlying stock is P n -distributed. Under which conditions does a subsequence of F (n) 0 converge to an arbitrage free price F 0 of that derivative of a P-distributed stock? We will first construct examples that show, that even if F (n) 0 and F 0 are uniquely determined (as in the log-binomial and log-normal model), subsequences of F (n) 0 do not necessarily converge to F 0 . On the other hand we will observe that these examples allow &quot;asymptotically arbitrages&quot;, i.e. strate..

ON MEAN ERGODIC CONVERGENCE IN THE CALKIN ALGEBRAS

(Communicated by) Abstract. In this paper, we give a geometric characterization of mean ergodic convergence in the Calkin algebras for Banach spaces that have the bounded compact approximation property. 1