Avodah Zarah, Daf Kaf Aleph, Part 5
ובכל מקום לא ישכור וכו’: תניא רבן שמעון בן גמליאל אומר לא ישכור אדם מרחצו לעובד כוכבים מפני שנקרא על שמו ועובד כוכבים זה עושה בו מלאכה בשבתות ובימים טובים
In all places one should not rent them a bath-house. It has been taught: Rabban Shimon b. Gamaliel said: One should not rent his bath-house to a non-Jew, for it is called by the owner’s name, and the non-Jew will work in it on Shabbat and festivals.
One should not rent a bath-house to a non-Jew because this will look as if they are performing work on behalf of the Jew on Shabbat.
אבל לכותי מאי שרי כותי אימר עביד ביה מלאכה בחולו של מועד
But to a Samaritan is it permitted? Say that he might do work on the intermediate days of the festival.
By implication, renting a bath-house to a Samaritan should be permitted. But while Samaritans observe the Sabbath and do not work on it, they do work on intermediate days of the festival. The assumption here is that Jews do not work on these days and thus people might think that he is working on behalf of the Jew.
בחולו של מועד אנן נמי עבדינן.
On the intermediate days we too do such work.
The answer is that Jews also are allowed to work in a bath-house on hol hamoed by heating up the water.
אבל שדהו לעובד כוכבים מאי שרי? מאי טעמא אריסא אריסותיה קעביד מרחץ נמי אמרי אריסא אריסותיה קעביד
אריסא דמרחץ לא עבדי אנשי
But renting a field to a non-Jew is permitted! What is the reason? Because people will say that he is merely a tenant working for his tenancy.
With regard to a bath-house people will also say that he is working for his tenancy?
People do not generally rent a bath-house on terms for tenancy.
A Jew is allowed to rent, at least outside of Israel, a field to a non-Jew, even though non-Jews will certainly work there on Shabbat. Why are we not concerned lest people think that non-Jews are working for Jews on Shabbat? The answer is that people will think that he is a tenant someone working for a share of the crops. It is permitted to have such an arrangement.
But then why not say the same thing about the bath-house people will think that the non-Jew has a tenant relationship with the Jew, and works the bath-house for a share of the profits? The answer is that such business relationships are not typical for bath-houses.