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Ergodic theorem for the algebra of operators in \(\mathbb L^p\). (English) Zbl 1155.37002

Let \((X,\mathcal B,\mu,T)\) be a measure-preserving system on a probability space, and fix \(p\geq1\). Then \(\widetilde{T}A=U_TAU_T^{-1}\), \(U_Tf=f\circ T\) defines an isometric algebraic automorphism of the algebra \(B_p\) of bounded linear operators on \(L^p(\mu)\). The automorphism \(\widetilde{T}\) does not have a subinvariant faithful normal state on \(B_2\), and this gives rise to essential differences between the ergodic theory of \(\widetilde{T}\) and that of positive maps on a von Neumann algebra. In order to develop ergodic properties of \(\widetilde{T}\), it is restricted to \(C_p\), the subalgebra of compact operators, and here the following ergodic theorem is found. If \(T\) is weak-mixing and \(p\geq1\), then \((1/N)\sum_{k=1}^{N}(\widetilde{T}^kA)g\rightarrow\mathbb{E}(A^*1)\cdot\mathbb{E}(g)\) in \(L^1(\mu)\) for any \(A\in C_p\) and \(g\in L^p(\mu)\). The proof uses a generalized Haar system to decompose \(L^p(\mu)\).

MSC:

37A05 Dynamical aspects of measure-preserving transformations
37A30 Ergodic theorems, spectral theory, Markov operators
37A25 Ergodicity, mixing, rates of mixing
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References:

[1] Jajte, Strong Limit Theorems in Non-Commutative Probability (1985) · doi:10.1007/BFb0101453
[2] Paulsen, Completely Bounded Maps and Operator Algebras (2003) · Zbl 1029.47003 · doi:10.1017/CBO9780511546631
[3] Pisier, Introduction to Operator Space Theory (2003) · Zbl 1093.46001 · doi:10.1017/CBO9781107360235
[4] Halmos, Lectures on Ergodic Theory (1956)
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