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On automorphisms of \(C^{*}\)-algebras whose Voiculescu entropy is genuinely non-commutative. (English) Zbl 1235.37010
Let \(A\) be an (exact) \(C^*\)-algebra, \(\alpha \in \text{Aut}(A)\) and let \(ht(\alpha)\) be the Voiculescu topological entropy of \(\alpha\). Set \(ht_C(\alpha)= \sup \{ht(\alpha |_C): C \text{ a commutative }\alpha\text{-invariant } C^*\text{-subalgebra of } A\}\). It is shown that the bitstream shifts studied in [S. Neshveyev and E. Størmer, Dynamical entropy in operator algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 50. Berlin: Springer (2006; Zbl 1109.46002)] give counterexamples to the equality \(ht(\alpha)=ht_C(\alpha)\).
MSC:
37B40 Topological entropy
46L55 Noncommutative dynamical systems
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[1] DOI: 10.1017/S0143385700009718 · Zbl 0832.46059 · doi:10.1017/S0143385700009718
[2] Neshveyev, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) (2006)
[3] DOI: 10.1007/PL00005539 · Zbl 0998.46033 · doi:10.1007/PL00005539
[4] DOI: 10.1007/s002220050229 · Zbl 0919.46043 · doi:10.1007/s002220050229
[5] DOI: 10.1007/s11005-008-0279-y · Zbl 1182.46055 · doi:10.1007/s11005-008-0279-y
[6] DOI: 10.1006/jfan.2000.3705 · Zbl 0992.46052 · doi:10.1006/jfan.2000.3705
[7] DOI: 10.1007/BF01225381 · Zbl 0637.46073 · doi:10.1007/BF01225381
[8] DOI: 10.1007/BF02392643 · Zbl 1021.46050 · doi:10.1007/BF02392643
[9] DOI: 10.1007/BF02108329 · Zbl 0824.46079 · doi:10.1007/BF02108329
[10] DOI: 10.1017/S0143385798108325 · Zbl 0932.46057 · doi:10.1017/S0143385798108325
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